The accurate measurement of the orientation of optical parts and systems is a pressing problem for upcoming space missions, such as stellar interferometers, requiring the knowledge and maintenance of positions to the sub-arcsecond level. Theodolites, the devices commonly used to make these measurements, cannot provide the needed level of accuracy. This paper describes the design, construction, and testing of an interferometer system to fill the widening gap between future requirements and current capabilities. A Twyman-Green interferometer mounted on a 2 degree of freedom rotation stage is able to obtain sub-arcsecond, gravity-referenced tilt measurements of a sample alignment cube. Dubbed a 'theoferometer,' this device offers greater ease-of-use, accuracy, and repeatability than conventional methods, making it a suitable 21st-century replacement for the theodolite.